Abstract
Let M n = X 1 + ⋯ + X n be a sum of independent random variables such that X k ⩽ 1, \(\mathbb{E}X_k = 0\) and EX 2 k = σ 2 k for all k. Hoeffding [15, Theorem 3] proved that
with
. Bentkus [5] improved Hoeffding’s inequalities using binomial tails as upper bounds. Let \(\gamma _k = \mathbb{E}{{X_k^3 } \mathord{\left/ {\vphantom {{X_k^3 } {\sigma _k^3 }}} \right. \kern-\nulldelimiterspace} {\sigma _k^3 }}\) and \(\kappa _k = \mathbb{E}{{X_k^4 } \mathord{\left/ {\vphantom {{X_k^4 } {\sigma _k^4 }}} \right. \kern-\nulldelimiterspace} {\sigma _k^4 }}\) stand for the skewness and kurtosis of X k . In this paper we prove (improved) counterparts of the Hoeffding inequality replacing σ 2 by certain functions of γ 1, ..., γ n (respectively ϰ1, ..., ϰ1). Our bounds extend to a general setting where X k are martingale differences, and they can combine the knowledge of skewness and/or kurtosis and/or variances of X k . Up to factors bounded by e 2/2 the bounds are final. All our results are new since no inequalities incorporating skewness or kurtosis control are known so far.
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References
V. Bentkus, An inequality for large deviation probabilities of sums of bounded i.i.d. random variables, Lith. Math. J., 41(2):112–119, 2001.
V. Bentkus, An inequality for tail probabilities of martingales with bounded differences, Lith. Math. J., 42(3):255–261, 2002.
V. Bentkus, A remark on the inequalities of Bernstein, Prokhorov, Bennett, Hoeffding, and Talagrand, Lith. Math. J., 42(3):262–269, 2002.
V. Bentkus, An inequality for tail probabilities of martingales with differences bounded from one side, J. Theor. Probab., 16(1):161–173, 2003.
V. Bentkus, On Hoeffding’s inequalities, Ann. Probab., 158(2):1650–1673, 2004.
V. Bentkus, On measure concentration for separately Lipschitz functions in product spaces, Israel. J. Math., 158:1–17, 2007.
V. Bentkus, G.D.C. Geuze, M.G.F. Pinenberg, and M. van Zuijlen, Unimodality: The symmetric case, Report No. 0612 of Dept. of Math. Radboud University Nijmegen, pp. 1–12, 2006.
V. Bentkus, G.D.C. Geuze, and M. van Zuijlen, Optimal Hoeffding-like inequalities under a symmetry assumption, Statistics, 40(2):159–164, 2006.
V. Bentkus, G.D.C. Geuze, and M. van Zuijlen, Unimodality: The general case, Report No. 0608 of Dept. of Math. Radboud University Nijmegen, pp. 1–24, 2006.
V. Bentkus, G.D.C. Geuze, and M. van Zuijlen, Unimodality: The linear case, Report No. 0607 of Dept. of Math. Radboud University Nijmegen, pp. 1–11, 2006.
V. Bentkus, N. Kalosha, and M. van Zuijlen, On domination of tail probabilities of (super)martingales: explicit bounds, Lith. Math. J., 46(1):3–54, 2006.
V. Bentkus, N. Kalosha, and M. van Zuijlen, Confidence bounds for the mean in nonparametric multisample problems, Stat. Neerl., 61(2):209–231, 2007.
V. Bentkus and M. van Zuijlen, On conservative confidence intervals, Lith. Math. J., 43(2):141–160, 2003.
A. Godbole and P. Hitczenko, Beyond the method of bounded differences, Microsurveys in Discrete Probability, 41:43–58, 1998. (Princeton, NJ, 1997), Dimacs Ser. Discrete Math. Theor. Comput. Sci., Am. Math. Soc., Providence, RI.
W. Hoeffding, Probability inequalities for sums of bounded random variables, J. Am. Stat. Assoc., 58:13–30, 1963.
N. Laib, Exponential-type inequalities for martingale difference sequences. application to nonparametric regression estimation, Commun. Stat. Theory Methods, 28:1565–1576, 1999.
C. McDiarmid, On the method of bounded differences, in Surveys in Combinatorics, Norwich, London Math. Soc. Lecture Note Ser., 1989, pp. 148–188.
F. Perron, Extremal properties of sums of Bernoulli random variables, Stat. Probab. Lett., 62: 345–354, 2003.
I. Pinelis, Optimal tail comparison based on comparison of moments, in High Dimensional Probability, Progr. Probab., 43, Birkhauser, Basel, Oberwolfach, 1998, pp. 297–314.
I. Pinelis, Fractional sums and integrals of r-concave tails and applications to comparison probability inequalities, in Advances in Stochastic Inequalities, Contemp. Math., 234, Am. Math. Soc., Providence, RI, Atlanta, GA, 1999, pp. 149–168.
I. Pinelis, On normal domination of (super)martingales, Electron. J. Prabab., 11(39): 1049–1070, 2006.
I. Pinelis, Inequalities for sums of asymmetric random variables, with applications, Probab. Theory Relat. Fields, 139(3–4):605–635, 2007.
I. Pinelis, Toward the best constant factor for the rademacher-gaussian tail comparison, ESAIM Probab. Stat., 11:412–426, 2007.
M. Talagrand, The missing factor in hoeffding’s inequalities, Ann. Inst. H. Poincaré Probab. Stat., 31(4):689–702, 1995.
S.A. van de Geer, On hoeffding’s inequalities for dependent random variables, in Empirical Process Techniques for Dependent Data, Contemp. Math., 234, Am. Math. Soc., Providence, RI, Birkhauser Boston, Boston, MA, 2002, pp. 161–169.
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The research was partially supported by the Lithuanian State Science and Studies Foundation, grant No T-15/07.
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Bentkus, V., Juškevičius, T. Bounds for tail probabilities of martingales using skewness and kurtosis. Lith Math J 48, 30–37 (2008). https://doi.org/10.1007/s10986-008-0003-8
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DOI: https://doi.org/10.1007/s10986-008-0003-8